We saw that El-Gamal encryption (Eq. (1) may be re-randomized, using the recipients public key, and mentioned that this may be extended into an encryption scheme which is univerally re-randomizable, i.e. where re-randomization does not require the recipients public key. Design such encryption scheme. Hint: begin with El-Gamal encryption, and use as part of the ciphertext, the result of encrypting the number 1.

c = m (g^a)^r mod p , v = g^r mod p (1)